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Full text of "Mathematical And Physical Papers - Iii"

I                        .
J                              20    ON THE EFFECT  OF  THE  INTERNAL  FRICTION  OF FLUIDS
I                         Since JST2 must not become infinite when # = oo, we must have
(7 = 0, D = 0.    Obtaining X% from the first of equations (11), and j                         substituting in (10), we get
!  !                                                in     r               f             I n      \
; .<                          fl = e~vv \ — Asm[nt—./^—,^} + Bco&\
I                                                       [              V      V 2/z   y
Now by the equations of conditions assumed in Art. 3, we must 5 !                            have V = V when cc = 0, whence

-    /^,
To find the normal and tangential components of the pressure of the fluid on the plane, we must substitute the above value of v in the formulae (4), (5), and after differentiation put x = 0. PI} TB will then be the components of the pressure of the solid on the fluid, and therefore — P19 — T9, those of the pressure of the fluid on the solid. We get
The force expressed by the first of these terms tends to diminish the amplitude of the oscillations of the plane. The force expressed by the second has the same effect as increasing the inertia of the plane.
8. The equation (12) shews that a given phase of vibration is propagated from the plane into .the fluid with a velocity \/(2///n), while the amplitude of oscillation decreases in geometric progression as the distance from the plane increases in arithmetic. If we suppose the time of oscillation from rest to rest to be one second, n = 7r; and if we suppose vV — 'HS inch, which, as will presently be seen, is about its value in the case of air, we get for the velocity of propagation '2908 inch per second nearly. If we enquire the distance from the plane at which the amplitude of oscillation is reduced to one-half, we have only to put
which gives, on the same suppositions as before respecting numerical values, cc= -06415 inch nearly. For water the value of //,' is a good deal smaller than for air, and the corresponding value of x smaller likewise, since it varies cceteris paribus as *Jp!. Hence if